CVFEM for influence of external magnetic source on Fe3O4-H2O nanofluid behavior in a permeable cavity considering shape effect

International Journal of Heat and Mass Transfer 115 (2017) 180–191

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CVFEM for influence of external magnetic source on Fe3O4-H2O nanofluid behavior in a permeable cavity considering shape effect M. Sheikholeslami a,⇑, S.A. Shehzad b,⇑ a b

Department of Mechanical Engineering, Babol Noshiravni University of Technology, Babol, Iran Department of Mathematics, COMSATS Institute of Information Technology, Sahiwal 57000, Pakistan

a r t i c l e

i n f o

Article history: Received 10 June 2017 Received in revised form 9 July 2017 Accepted 10 July 2017

Keywords: Nanofluid External magnetic source Shape effect Porous media MFD viscosity Natural convection

a b s t r a c t This research is made to show the impact of external magnetic source on Fe3O4 – water nanofluid treatment in a permeable cavity. Shape factor effect on nanofluid properties are taken into account. Final equations are derived by means of vorticity stream function formulation. Control volume based finite element method is employed for solving final formulae. Figures are depicted for various values of radiation parameter, Darcy number, Fe3O4 –water volume fraction, Rayleigh number and Hartmann number. Results reveal that selecting Platelet shaped nanoparticles results the highest heat transfer rate. Nanofluid velocity and heat transfer rate decrease with augment of Hartmann number. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction In order to improve heat transfer rate, several ways offered. One of them is Nanotechnology. Rashidi et al. [1] reported the effect of Lorentz forces on nanofluid mixed convection in a wavy channel. Sheikholeslami and Rokni [2] simulated the impact of melting heat transfer on nanofluid flow in existence of magnetic field. Beg et al. [3] reported the bio-nanofluid transportation by means of both nanofluid models. Sheikholeslami and Shehzad [4] demonstrated the nanofluid natural convection in a permeable media by means LBM. Garoosi et al. [5] utilized the two phase model for nanofluid flow in presence of external heating. Sheikholeslami and Shehzad [6] reported the influence of radiation on ferrofluid flow. They considered MFD viscosity. Sheikholeslami and Bhatti [7] demonstrated the effect of nanoparticles shape on nanofluid forced convection. Khan et al. [8] investigated the influence of heat generation on transient nanofluid flow on a wedge. Garoosi et al. [9] reported the application of nanofluid in a heat exchanger. Their result showed that situation of the hot tube has sensible effect on temperature. Sheikholeslami and Bhatti [10] utilized active method for heat transfer augmentation in an enclosure filled with nanofluid. Mesoscopic approach has been used by Sheikholeslami and Ellahi [11] for a three dimensional problem. ⇑ Corresponding authors. E-mail addresses: [email protected] (M. Sheikholeslami), [email protected] (S.A. Shehzad). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2017.07.045 0017-9310/Ó 2017 Elsevier Ltd. All rights reserved.

Feng et al. [12] presented a numerical simulation for melting of nanofluid in a cavity. Sheikholeslami and Rashidi [13] investigated nanofluid convective heat transfer in existence of variable magnetic field. Impact of variable Kelvin forces on magnetic nanofluid flow has been simulated by Sheikholeslami Kandelousi [14]. Heat flux boundary condition has been utilized by Sheikholeslami and Shehzad [15] to investigate the magnetic nanofluid flow in porous media. Nanoparticle transportation in a channel in presence of Lorentz forces was demonstrated by Akbar et al. [16]. Sheikholeslami et al. [17] studied the impact of radiation on distribution of nanofluid. Sheikholeslami and Zeeshan [18] employed mesoscopic method for CuO-water nanofluid flow in a porous cavity. In recent years, different researchers published articles about nanofluid flow [19–45]. This article deals with influence of non-uniform magnetic field on nanofluid convective flow inside an enclosure by means of CVFEM. Roles of Darcy number, radiation parameter, Fe3O4-water volume fraction, shape of nanoparticle, Hartmann and Rayleigh numbers are examined. 2. Problem statement Fig. 1 shows the sample element and boundary conditions of current geometry. As shown in Fig. 2, external magnetic source affect the nanofluid flow. H; Hx ; Hy are:

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Nomenclature K H V; U Da Ra T Nu Ha

X; W

Permeability Magnetic field Vertical and horizontal dimensionless velocity Darcy number Rayleigh number Fluid temperature Nusselt number Hartmann number

b

q r l

dimensionless vorticity & stream function Thermal expansion coefficient Fluid density Electrical conductivity Dynamic viscosity

Subscripts s Solid particle nf Nanofluid f Base fluid

Greek symbols h dimensionless temperature f Rotation angle

Fig. 1. (a) Geometry and the boundary conditions with; (b) A sample triangular element and its corresponding control volume.

Hy ¼ ða  xÞ Hx ¼ ðy  bÞ

1 2 c ½ðb  yÞ þ ða  xÞ2  ; 2p

c

2p

2

1

½ðb  yÞ þ ða  xÞ2  ;

0:5

H ¼ ðH2y þ H2x Þ :

ð1Þ ð2Þ ð3Þ

3. Governing equation and simulation 3.1. Governing formulation Convective nanofluid flow in existence of non-uniform magnetic field is considered using non-Darcy model. The PDEs are [30]:

@u @v ¼ ; @x @y

ð4Þ

! @2u @2u @P þ lnf   l20 rnf H
2y u þ rnf l20 H
x H
y v @y2 @x2 @x   lnf @u @u  v ; uþ u ¼ ðqnf Þ @x @y K !

lnf @P
y rnf H
x u  l2 H

þ l20 H 0 x rnf H x v  @y K   @v @v þ ðT  T c Þbnf g qnf ¼ qnf v ; uþ @x @y

lnf

@2v @2v þ @x2 @y2

ð5Þ



v

ð6Þ

182

M. Sheikholeslami, S.A. Shehzad / International Journal of Heat and Mass Transfer 115 (2017) 180–191

9 8 7 6 5 4 3 2 1 -1 -2 -3 -4 -5 -6 -7 -8 -9

19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2

(a) H ( x , y )

(b) Hx ( x , y )

-1 -2 -3 -4 -5 -6 -7 -8 -9 -10 -11 -12 -13 -14 -15 -16 -17 -18 -19

(c) Hy ( x , y ) Fig. 2. Contours of the (a) magnetic field strength H; (b) magnetic field intensity component in x direction Hx; (c) magnetic field intensity component in y direction Hy.

knf

@2T @2T þ @y2 @x2

! 

@qr ¼ @y



v

 @T @T ðqC p Þnf ; þu @y @x

4re @T 4 ; T 4 ffi 4T 3c T  3T 4c : ½qr ¼  3bR @y ðqbÞnf ; ðqC p Þnf , qnf and

qnf ¼ qf ð1  /Þ þ qs / ð7Þ

rnf ¼ rf



 3/ðr1  1Þ þ1 ; ð1  r1Þ/ þ ð2 þ r1Þ

ð10Þ

r1 ¼ rs =rf :

ð11Þ

ln f is calculate as [46]:

rnf are calculated as:

ðqbÞnf ¼ ðqbÞf ð1  /Þ þ ðqbÞs /;

ð8Þ

ðqC p Þnf ¼ ðqC p Þf ð1  /Þ þ ðqC p Þs /

ð9Þ

lnf ¼ ð0:035l20 H
2 þ 3:1l0 H
 27886:4807/2 þ 4263:02/ þ 316:0629Þe0:01T knf can be obtained as [19]:

ð12Þ

183

M. Sheikholeslami, S.A. Shehzad / International Journal of Heat and Mass Transfer 115 (2017) 180–191 Table 1 Thermo physical properties of water and nanoparticles.

Pure water Fe3 O4

qðkg=m3 Þ

C p ðj=kgKÞ

kðW=mkÞ

dp ðnmÞ

rðX  mÞ1

997.1 5200

4179 670

0.613 6

– 47

0:05 25000

Table 2 The values of shape factor of different shapes of nanoparticles [24]. m

Spherical

3

Platelet

5.7

Cylinder

4.8

Brick

3.7

knf mðkf  kp Þ/ þ ðkp  kf Þ/ þ mkf þ kp þ kf ¼ kf mkf þ ðkf  kp Þ/ þ kf þ kp

ð13Þ

Tables 1 and 2 show the properties of nanofluid and shape factor values. Vorticity and stream function can be employed to eliminate pressure source terms:

xþ

@u @ v @w @w  ¼ v ; ¼ u: ¼ 0; @y @x @x @y

ð14Þ Fig. 3. Comparison of temperature profile between the present results and numerical results by Khanafer et al. [49] Gr ¼ 104 , / ¼ 0:1 and Pr ¼ 6:8ðCu  WaterÞ.

Dimensionless parameters are defined as:

ðb; aÞ ; L uL vL ðx; yÞ T  Tc ;h ¼ ;V ¼ ; ðX; YÞ ¼ ; DT ¼ q00 L=kf ; U¼ L anf anf DT

ðHy ; Hx ; HÞ ¼

ðHy ; Hx ; HÞ H0

; ðb; aÞ ¼

ð15Þ

xL2 W¼ ;X ¼ : anf anf w

@Y 2

þXþ

@2W @X 2

¼ 0;

ð16Þ

! @X @X A5 A2 @ 2 X @ 2 X þ UþV ¼ Pr @X @Y A1 A4 @Y 2 @X 2   A6 A2 @U @V 2 @U 2 @V Hy Hx  Hx þ Hy  H y Hx þ PrHa2 A1 A4 @X @X @Y @Y þ Pr Ra

on

Nusselt

number

when

Ha

So equations change to:

@2W

Table 5 Effect of shape of nanoparticles Da ¼ 100; Ra ¼ 105 ; Rd ¼ 0:8; / ¼ 0:04.

ð17Þ

Spherical Brick Cylinder Platelet

@h @h UþV ¼ @X @Y

@2h @X

2

þ

@2h @Y

0

10

7.072375 7.120273 7.196075 7.258503

5.865269 5.901963 5.960649 6.009695

!

2

þ

4 1 @2h Rd 2 : 3 A4 @Y

ð18Þ

and dimensionless parameters are

A3 A22 @h Pr A5 A2  ; A1 A24 @X Da A1 A4

Table 3 Comparison of Nuav e along curved wall for different grid resolution at Ra ¼ 105 , Da ¼ 100; / ¼ 0:04; Rd ¼ 0:8, Ha ¼ 10; Ec ¼ 105 and Pr ¼ 6:8. 51  151

61  181

71  211

81  241

91  271

101  301

6.007844

6.008708

6.009695

6.010115

6.012651

6.016738

Table 4 Average Nusselt number versus at different Grashof number under various strengths of the magnetic field at P r = 0.733. Ha

0 10 50 100

Gr ¼ 2  105

Gr ¼ 2  104 Present

Rudraiah et al. [48]

Present

Rudraiah et al. [48]

2.5665 2.26626 1.09954 1.02218

2.5188 2.2234 1.0856 1.011

5.093205 4.9047 2.67911 1.46048

4.9198 4.8053 2.8442 1.4317

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-2

-8 -14 0.15 -18

0.2

0.1

0.25

0.05

Fig. 4. Impact of nanofluid volume fraction on streamlines (right) and isotherms (left) contours (nanofluid (/ ¼ 0:04)(––) and pure fluid(/ ¼ 0) (  )) when Ra ¼ 105 ; Da ¼ 100; Rd ¼ 0:8; Ha ¼ 0.

-4

-1 0 0.2

-18

0.15

-18 0.1 0.25 0. 35

0.05

Fig. 5. Impact of radiation parameter on streamlines (right) and isotherms (left) contours (Rd ¼ 0:8 (––), Rd ¼ 0 (—)) when Ra ¼ 105 ; Da ¼ 100; / ¼ 0:04; Ha ¼ 0.

Raf ¼ gbf L3 DT=ðaf tf Þ; Prf ¼ tf =af ; Ha ¼ Ll0 H0

qffiffiffiffiffiffiffiffiffiffiffiffiffi rf =lf ;

and boundary conditions are:

Da ¼ K= L2 ; Ec ¼ ðlf af Þ=½ðqC P Þf DT L2 ;

ðqC p Þnf ðqbÞnf qnf ; A2 ¼ ; A3 ¼ ; qf ðqC p Þf ðqbÞf lnf knf rnf A4 ¼ ; A5 ¼ ;A ¼ ; Rd ¼ 4re T 3c =ðbR kf Þ kf lf 6 rf

A1 ¼

@h ¼ 1:0 @n on outer wall h ¼ 0:0 @h on other walls ¼0 @n on all walls W ¼ 0:0 on inner wall

ð19Þ

ð20Þ

Ha=0

0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.15 0.05

Ha=10

Da=100

Ha=0

Da=0.01

0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

Ha=10

M. Sheikholeslami, S.A. Shehzad / International Journal of Heat and Mass Transfer 115 (2017) 180–191

0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.65 0.6 0.5 0.4 0.3 0.25 0.2 0.15 0.1 0.05

-0.01 -0.02 -0.03 -0.04 -0.05 -0.06 -0.07 -0.08 -0.09 -0.1

-0.005 -0.01 -0.015 -0.02 -0.025 -0.03 -0.035 -0.04 -0.045 -0.05 -0.055 -0.06

-0.05 -0.1 -0.15 -0.2 -0.25 -0.3 -0.35 -0.4

-0.01 -0.02 -0.03 -0.04 -0.05 -0.06 -0.07 -0.08 -0.09 -0.1 -0.11 -0.12

Fig. 6. Influence of Da; Ha on streamlines (right) and isotherms (left) contours when / ¼ 0:04; Ra ¼ 103 .

185

Ha=0

0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

-0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8 -0.9 -1 -1.1

Ha=0

-0.05 -0.1 -0.15 -0.2 -0.25 -0.3 -0.35 -0.4 -0.45 -0.5 -0.55 -0.6 -0.65

0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

-0.5 -1 -1.5 -2 -2.5 -3 -3.5 -4 -4.5 -5 -5.5

Ha=10

Da=0.01

0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

Ha=10

M. Sheikholeslami, S.A. Shehzad / International Journal of Heat and Mass Transfer 115 (2017) 180–191

0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

-0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8 -0.9 -1 -1.1 -1.2 -1.3 -1.4 -1.5

Da=100

186

Fig. 7. Influence of Da; Ha on streamlines (right) and isotherms (left) contours when / ¼ 0:04; Ra ¼ 104 .

0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

-1 -2 -3 -4 -5 -6 -7 -8 -9 -10

0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

-0.5 -1 -1.5 -2 -2.5 -3 -3.5 -4 -4.5 -5 -5.5 -6 -6.5 -7 -7.5 -8

Ha=0

-2 -4 -6 -8 -10 -12 -14 -16 -18

0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

-1 -2 -3 -4 -5 -6 -7 -8 -9 -10 -11 -12

Da=100

0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

Ha=10

Ha=10

Da=0.01

Ha=0

M. Sheikholeslami, S.A. Shehzad / International Journal of Heat and Mass Transfer 115 (2017) 180–191

Fig. 8. Influence of Da; Ha on streamlines (right) and isotherms (left) contours when / ¼ 0:04; Ra ¼ 105 .

187

188

M. Sheikholeslami, S.A. Shehzad / International Journal of Heat and Mass Transfer 115 (2017) 180–191

Ha = 5 , Da = 50 ,φ = 0.04

Ha = 5 , Ra = 10 4 ,φ = 0.04

Ra = 10 4 , Da = 50 ,φ = 0.04

Rd = 0.4 , Ha = 5 , φ = 0.04

Rd = 0.4 , Ra = 10 4 , φ = 0.04

Rd = 0.4 , Da = 50 , φ = 0.04

Fig. 9. Effects of Da; Ha and Ra on average Nusselt number.

Nuloc ; Nuav e over the hot wall can be calculated as:

Nuloc

!    1 knf knf 4Rd 1 1þ ; ¼ 3 h kf kf

Nuav e ¼

1 S

Z

ð21Þ

S

Nuloc ds: 0

3.2. Numerical procedure

ð22Þ

Benefits of both finite element and finite volume methods are combined in CVFEM. This method utilizes triangular element (see Fig. 1(b)). Upwind approach is utilized for advection term. GaussSeidel approach is applied to solve the algebraic equations. More details can be found in [47].

M. Sheikholeslami, S.A. Shehzad / International Journal of Heat and Mass Transfer 115 (2017) 180–191

Ha = 5 , Da = 50 ,φ = 0.04

Ha = 5 , Ra = 10 4 , φ = 0.04

Ra = 10 4 , Da = 50 ,φ = 0.04

Rd = 0.4 , Ha = 5 ,φ = 0.04

Rd = 0.4 , Da = 50 ,φ = 0.04

189

Rd = 0.4 , Ra = 10 4 ,φ = 0.04 Fig. 9 (continued)

4. Grid independency and code accuracy

5. Results and discussion

Various grids should be examined to mesh independency analysis. According to Table 3, a mesh size of 81  241 can be selected. This FORTRAN code has good accuracy for MHD flow as demonstrated in Table 4 [48]. The correctness of CVFEM code for nanofluid simulation is shown in Fig. 3 [49].

Effect of external magnetic source on Fe3O4-water hydrothermal treatment in a permeable enclosure is demonstrated. Magnetic field dependent viscosity of nanofluid is considered. Shape effect of nanoparticles on knf is taken into consideration. CVFEM is employed to find the influences of Darcy number (Da ¼ 0:01 to

190

100),

M. Sheikholeslami, S.A. Shehzad / International Journal of Heat and Mass Transfer 115 (2017) 180–191

radiation

parameter

ðRd ¼ 0 to 0:8Þ,

Rayleigh

number

(Ra ¼ 103 ; 104 ; 105 ), volume fraction of Fe3O4 (/ ¼ 0% to 4%), shape factor and Hartmann number (Ha ¼ 0 to 10). Effect of shape factor on Nuav e is discussed in table 5. The highest Nuav e is caused by Platelet, followed by Cylinder, Brick and Spherical. So, Platelet shape was selected for further investigation. Fig. 4 illustrates the influence of adding nanoparticles in the base fluid on velocity and temperature contours. Temperature gradient reduces with augment of /. jWmax j increases with adding nanoparticles into the base fluid. Fig. 5 shows the impact of radiation parameter on isotherms and streamlines. Increasing radiation parameter leads to augment in thermal boundary layer thickness. In existence of Lorentz forces, impact of radiation parameter on flow style becomes no sensible. Figs. 6, 7 and 8 depicts the influence of Da; Ra; Ha; Rd on isotherms and streamlines. In conduction mode (low Reynolds number), streamline shows one rotating eddy. In existence of magnetic field, the main eddy moves downward. The distortion of isotherms becomes less than before. As buoyancy forces enhances, thermal plume generates. As Darcy number enhances, convective mode becomes stronger and rate of heat transfer enhances. Effect of significant parameters on Nuav e is demonstrated in Fig. 9. The formula for Nuav e is:

Nuav e ¼ 15:28  1:12Rd  6:6 logðRaÞ  0:18Da þ 0:47Ha þ 1:07Rd logðRaÞ þ 0:32Rd Da  0:44Rd Ha

ð23Þ

þ 0:08Da logðRaÞ  0:11Ha logðRaÞ  0:3Da Ha  0:9Rd þ 0:85ðlogðRaÞÞ þ 0:15ðDa Þ  0:07ðHa Þ 2

2

2

2

where Da ¼ 0:01Da; Ha ¼ 0:1Ha. Nusselt enhances with augment of buoyancy forces and permeability of porous media. Temperature gradient reduces with increase of Hartmann number. Also, Nusselt number enhances with rise of radiation number. 6. Conclusions Effect of external magnetic source on magnetic nanofluid behavior in a porous enclosure has been presented. Shape factor and magnetic field effects on thermal conductivity and viscosity of nanofluid are taken into consideration. Temperature distribution and flow style are reported for different values of radiation parameter, Hartmann, Darcy and Rayleigh numbers. Outputs illustrate that choosing Platelet shape of nanoparticles leads to greatest Nusselt number. Furthermore, Nusselt number reduces with augment of Lorentz forces. Darcy and Rayleigh numbers can enhance the temperature gradient. References [1] M.M. Rashidi, M. Nasiri, M. Khezerloo, N. Laraqi, Numerical investigation of magnetic field effect on mixed convection heat transfer of nanofluid in a channel with sinusoidal walls, J. Magnet. Magnet. Mater. 401 (2016) 159–168. [2] M. Sheikholeslami, H.B. Rokni, Melting heat transfer influence on nanofluid flow inside a cavity in existence of magnetic field, Int. J. Heat Mass Transf. 114 (2017) 517–526. [3] O.A. Beg, M.M. Rashidi, M. Akbari, A. Hosseini, Comparative numerical study of single-phase and two-phase models for bio-nanofluid transport phenomena, J. Mech. Med. Biol. 14 (2014) 1450011. [4] M. Sheikholeslami, S.A. Shehzad, Magnetohydrodynamic nanofluid convective flow in a porous enclosure by means of LBM, Int. J. Heat Mass Transf. 113 (2017) 796–805. [5] F. Garoosi, B. Rohani, M.M. Rashidi, Two-phase mixture modeling of mixed convection of nanofluids in a square cavity with internal and external heating, Powder Tech. 275 (2015) 304–321. [6] M. Sheikholeslami, S.A. Shehzad, Thermal radiation of ferrofluid in existence of Lorentz forces considering variable viscosity, Int. J. Heat Mass Transf. 109 (2017) 82–92. [7] M. Sheikholeslami, M.M. Bhatti, Forced convection of nanofluid in presence of constant magnetic field considering shape effects of nanoparticles, Int. J. Heat Mass Transf. 111 (2017) 1039–1049.

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